In 2004 William Dembski disseminated an article wherein he introduced an
allegedly novel measure of information he dubbed “Variational Information.”
This measure was in fact not a novel quantity but rather a well known (for
over forty years) “Rényi divergence of the 2nd order.” When his
flop was pointed out, Dembski added to his article a reference to Rényi.
However, the amended version of Dembski’s article contained statements which
showed that Dembski not only was not familiar with the widely known and
heavily referenced work of Rényi, but did not grasp its essence even after
it was pointed out to him. In the three years after the amended
version of Dembski’s article was posted, Dembski has kept that version
posted without any signs of having a second thought. This allows one to
think that no self-correction from Dembski is forthcoming. These facts seem
to support the opinion of Dembski as being, in the words of Professor
Shallit, a “pseudo-mathematician”. I similarly suggested that his
mathematical exercises are “quasi-mathematics.” Contrary to Dembski’s
claims, it in no way can serve as a “mathematical foundation of Intelligent
Design.” Although in the opinion of ID advocates it contains innovative and
useful stuff, all that is “innovative” there is usually useless, while all
that is useful there has usually been known for a long time.

I believe it was a famous French philosopher who said
that everybody can make mistakes but only a fool stubbornly adheres to
them. The story I am going to tell now may serve as an illustration of the
above maxim, with a small modification: although the term “fool” may still
(arguably) reflect the situation, the term “incompetent” is applicable
unarguably. The hero of the story is William Dembski.

Professor Jeffrey O. Shallit is a well known
mathematician and computer scientist, the author of high level mathematical
monographs and articles in professional peer-reviewed journals. Naturally
when Jeffrey Shallit offers his opinion of the qualifications of authors of
publications in various areas of mathematics, it has considerable weight and
a good chance of being well thought of and strongly justified. This must be
especially true when such an opinion is about the mathematical prowess of
somebody who happened to be at some time in the recent past a student in a
class taught by Professor Shallit.

The case in point is that of a “leading light” of the
intelligent design movement, William A. Dembski, at present time reportedly
a research professor of philosophy of religion at a Baptist seminary in
Texas, who holds PhD degrees in both philosophy and mathematics. Dembski is
considered by ID advocates as the foremost mathematician within their
ranks. His colleagues and acolytes incessantly praise in superlative terms
Dembski’s contribution to ID.[1]

A double doctor (with additional degrees in psychology
and religious studies) is supposed to be an expert at least in the fields of
his educational specialization, in Dembski’s case including mathematics.
Oddly, Professor Shallit’s professional opinion of Dr-Dr. Dembski seems to
be rather unflattering. Indeed, in Dr. Shallit’s deposition preceding the
famous Dover trial, he unequivocally called Dembski a “pseudo-mathematician”
[2].

While I can’t speak for other readers of Shallit’s
deposition, I was not surprised in the least by his low opinion of Dembski
as a mathematician. Indeed, in Chapter 1 of my book [3] there is a section
wherein I showed Dembski’s penchant for loading his texts with esoteric
mathematical notations adding nothing to the gist of his discourse and
seeming to be a device applied solely for the sake of imbuing his output
with a quasi-mathematical appearance (the text of the section in question
can also be seen online [4]). It goes without saying that neither Dembski
nor any of his supporters has ever admitted the existence of my comments.

Given the high esteem in which Dembski is held in by ID advocates (as
illustrated by the references in [1]), they could no more be pleased by
Shallit’s characterization of Dembski’s mathematical exercises than by my
comments regarding his mathematically-empty notations-heavy passages.
Unfortunately for ID advocates, there is a shortage of mathematicians of
Shallit’s status among their “theorists.” Therefore they largely pretend
that Shallit’s characterization of Dembski’s mathematics does not exist, or
at the best let some of their second- and third-tier soldiers evince a
supposed rebuttal (often plainly meaningless) of Shallit’s work (as in the
case of Elsberry and Shallit’s devastating skewering of Dembski’s concept
of “Complex Specified Information” [5]).

It seems to be of a certain interest to look for other
evidence, independent of Professor Shallit’s or my opinion, which would
either support or contradict Shallit’s and my view of Dembski as a
mathematician. If Shallit and I are right, and the supposed mathematical
foundation of ID by Dembski is indeed of no value, this should reflect in a
strongly negative way upon the overall credibility of the intelligent design
conceptual system.

In view of the above, it seems to be instructive to
look at a story that took place in August 2004, when Dembski disseminated an
article wherein he claimed to have invented a novel measure of information.

In the original version of his paper (soon afterwards
removed from the web but not before it was downloaded and saved by a number
of readers) Dembski suggested a supposedly innovative measure of information
he dubbed “variational information” (VI).

Almost immediately, responding to Dembski,
mathematician Cosma R. Shalizi pointed out [6] that Dembski’s VI is nothing
more than a particular case of a well known (for over 40 years) quantity
known as Rényi divergence.

As a result of Shalizi’s comments, Dembski promptly
posted an amended version [7] (dated August 12, 2004) of his paper, where he
inserted references to Rényi .

Just in case that amended version of Dembski’s article
one day disappears from the above web location (like the original version
did), I have downloaded and saved it.

So far so good - a mathematician happened to “reinvent
the wheel;” such occurrences happen sometimes, and not only with
mathematicians – see for example an essay [8]. It was odd, though, that a
supposed expert in this particular field of mathematics was not familiar
with rather famous results by a prominent mathematician Alfréd Rényi, widely
published and heavily referenced for over forty years.

Since Dembski amended his paper, this could be the end
of the story, were it not for an additional quirk.

I am a physicist rather than a mathematician.
Nevertheless, I could not fail to notice that the amendment made by Dembski
in his article revealed that even after Shalizi pointed to Rényi’s result,
Dembski still seemed not only to be unfamiliar with an important result in
his supposed area of expertise, but also seemed to still simply not
comprehend Rényi’s work.

For example, in the Abstract that opens the amended
version of Dembski’s paper, we read, among other things, “The resulting
information measure is special case of the Rényi information divergence (also
known as the Rènyi entropy).” (emphasis added).

Let us look up definitions of Rényi entropy and of
Rényi divergence. Although in August 2004 I verified these definitions by
looking up Rényi’s own publications, today it seems to be easiest to look up
not Rényi’s original publications, but just Wikipedia [9].

Indeed, here they are:

Rényi divergence:

Rényi entropy is however

In the above equations " stands for the
order of either divergence or entropy; p and q are random
variables in two sets with different distributions.

(For continuous distributions these equations are
generalized by replacing sums with appropriate integrals).

Even for readers completely unfamiliar with Rényi’s
work, it is clear just by looking at the above equations that, contrary to
Dembski’s assertion cited above, Rènyi divergence is not “also known as
Rényi entropy,” because Rényi divergence and Rényi entropy are two different
quantities.

It looks like, even having accounted for Shalizi’s
critique and thus in fact implicitly admitting his insufficient ken in the
area of his supposed expertise, Dembski still has not done his homework
fully - he seems not to realize the difference between the above two
quantities, the difference illustrated even in such a commonly accessible
source as Wikipedia.

In the original version of his paper Dembski did not at
all refer to either of these two quantities, thus creating the impression
that he has introduced a novel type of information measure - variational
information. When Shalizi pointed out that “variational information” is
nothing more than the Rényi divergence of the order of 2 (i.e. when

"
=2), Dembski amended his paper but the above quotations from the amended
version seem to indicate that he still does not have an adequate grasp of
the matter.

Furthermore, in a response to Shalizi’s comments, which
was posted in 2004 at the ARN website, Dembski tried to downplay the initial
lack of references to Rényi . He still maintained that his paper contributed
something of substance to information theory. With his typical modesty,
Dembski did not hesitate to characterize his own alleged contribution as
“admirable.”

In particular, Dembski insisted that his paper was
innovative because allegedly nobody has ever used Rènyi’s measures of the
second order. In the amended paper he wrote, “The variational
information, though implicit as a special case in the Rényi information
divergence (r=2) has to date gone largely unnoticed and unappreciated.”
(Dembski notation r stands for " in the above
equations).

This assertion was not true. For example, in a paper by
López-Ruiz [10] it was shown that the quantity introduced by López-Ruiz,
Mancini, and Calbet [11] under the name of “disequilibrium” (which happens
to be a measure of a divergence of a probability distribution from
uniformity, thus functioning as another version of what Dembski suggested as
a supposedly novel measure of information) is closely related to the Rényi
entropy of the second order, via an exponential term.

Hence, there was little pioneering in Dembski’s
suggestion of using a second order version of Rènyi measures - it has been
done before. Again, Dembski obviously was not familiar with publications
directly related to his supposed area of expertise.

So it seems that the epithet “admirable,” suggested by
Dembski for his own paper, was not quite adequate for a reasonable
evaluation of his own work.

A natural question to ask is why have I waited over
three years before going public with the above critique? The answer is
simple: not to appear to be eager to pounce upon Dembski at each
opportunity; it seemed reasonable to give Dembski a chance to realize his
mistakes and to correct them on his own. In the three years since Dembski
published his error he has kept his “amended” article posted without any
signs of having realized his errors. This allows one to think that no
self-correction from Dembski is forthcoming.

This, in turn, allows one to assert that to all intents
and purposes Professor Shallit was right in defining Dembski as a
pseudo-mathematician and therefore that all the alleged “Mathematical
foundation of Intelligent design” so vigorously and triumphantly trumpeted
by Dembski and his acolytes, is pseudo-mathematics. It can’t serve as a
foundation of any reasonable conceptual set. Although in the opinion of ID
advocates it contains innovative and useful stuff, in fact all that is
“innovative” there is usually useless (like the confusion of Rényi
divergence with Rényi entropy), while all that is useful there has usually
been known for a long time (like Rényi divergence of the 2nd
order).

As has been pointed out more than once, by various
authors [12], the “theory” of intelligent design is scientifically empty.
Elementary errors, sometimes reluctantly admitted, but more often never
admitted at all, are expected even from the best educated ID practitioners
like William Dembski, not to mention his lesser admirers and followers.

REFERENCES AND NOTES

1. Statements claiming that William Dembski is a
first-rank mathematician are common in ID literature. Perhaps the most
exaggerated of these accolades was produced by an ID advocate, professor of
philosophy Rob Koons in a blurb to Dembski’s book “Intelligent Design”
(InterVarsity Press, 1999) where Koons referred to Dembski as “the Isaac
Newton of information theory,” and “one of the most important thinkers of
our time.” Michael Behe, in the foreword to the same book by Dembski,
claimed that the future development of science will be based on design and
hence on the “theoretical foundation of Bill Dembski’s work.” (page 12 in
Dembski’s book). Similar assertions can be found in the publications of many
other ID advocates.